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We consider the thermal relaxation of a particle in a piecewise-constantpotential landscape subject to thermal fluctuations in the overdamped limit. Westudy the connection between the occurrence of the Mpemba effect, the presenceof metastable states, and phase transitions as a function of the potential. Wefind that the Mpemba effect exists even in cases without metastable states. Inthe considered physical system, the borders of the areas where the effecthappens correspond to either eigenvector changes of direction or to phasetransitions. Finally, we discuss the topological aspects of the strong Mpembaeffect and propose using topology to search for the Mpemba effect in a physicalsystem.

Anomalous thermal relaxation of Langevin particles in a piecewise-constant potential